Tag Archives: calculus

Power and Maclaurin Series

23 Mar

I know my SIFE class is probably what more people want to hear about, but hey, it’s AP crunch time and Calc BC has been getting FUN. Finally, after 3 years of teaching it (and one year of suffering through it 17 years ago)!

Today’s share: I finally feel semi-decent about how I’m introducing the power and Maclaurin series. The last couple of years, my students have been full of complete panic when I introduced the first free response series examples because they looked so entirely different from the way the book introduced it. I tried looking at other books to try to supplement to see if they do a better job, but none were very satisfactory.

So I finally re-wrote it as group work for discovery. Let’s be real, after going through painstaking tests for series convergence and divergence for 8 sections straight, the power series should be an easy transition, but the fact is that it is NOT.

I think I’ve finally figured out why, or at least partly why. Firstly, the books kind of sneak in a second component to what an “interval of convergence” means for Maclaurin series. They quickly transition from using the “interval of convergence” to represent all values of x that make the series converge (like we’ve been doing for 8 sections) to suddenly using the phrase to also represent how two completely different functions are now graphically the same. Secondly, the books also fail to explicitly equate how the geometric series now uses x to represent the common ratio and now calls it a Maclaurin series. It is all very fuzzy and seemingly brand new, as opposed to being a continuation of the exact same concept we learned way back in the beginning of series.

So here is how I have introduced these two concepts, the power series to the Maclaurin series. Each of these worksheets are done in groups. I have different accountability measures for making sure each are doing the work together, but that is for a different post.

DAY 1 (95 min block): Worksheet A. This is to introduce the power series via the geometric series but by making the students explicitly point out the connection. I also explicitly point out the connection on which test to use using the same flow chart on the power series as well. Practice problems on using the Ratio Test on power series is given as classwork –> homework.

DAY 2 (55 min class): Worksheet B. This is where I try to redefine “interval of convergence” to mean something extra and introduce the Maclaurin series without actually introducing it. At the end of this day, w/ the catalyst of my cheesy/nerdy enthusiasm for how cool this is, I got quite a few “ohhhh. coooool!” from the kids. (FYI, this has never happened before in prior years.) No homework this day.

DAY 3 (95 min block): Worksheet C, pgs 1-2 first (and separately from the rest of the packet). I had them find the Maclaurin series for sine and cosine. At the end of these pages, have them MEMORIZE these function=series equivalents because otherwise, they keep wanting to re-derive every time!

While I am walking around checking, stamping, and collecting their pgs 1-2, they are memorizing. By the time I get to their desks to collect their pgs 1-2 I also have them clear their desks before handing them pgs 3-5. They now have to pull those series equivalence they just crammed out of their brains and on to paper.

I stop the class when about half the class has started the second part to go over an example of how to use our given four series to do their first “manipulating” of the four series together. Hw: finish finding the intervals of convergence and do the first three AP free response problems on pg 4. I encourage/ beg the students to please not Google the answers. They have a terrible “study habit” of having the answer too close and relying on it too much to find the answer.

DAY 4 (95 min block): Worksheet C, pgs 4-5. This day is yet to come for me (it is this Friday). I have plans to go over the answers in detail and the point distribution for the first three, then give group work time to go over the next three.

Taylor series and LaGrange Error Bound is yet to come. I have no clue how to introduce those yet. =/

 

Advertisements

Series Convergence/ Divergence

7 Mar

Super nerd alert. I have been working the last 3 hours on perfecting this flow chart that took me 3 years to figure out how best to go about proving series for convergence or divergence. If you have any idea what I’m talking about, I hope you find it useful. =)

(Please be aware that these are not perfect descriptions of the tests, only summaries. The second page is a fill in the blank copy. I think it would be fun to have a group competition to try to figure out every blank, then review/grade together as a class.)

screencapture

Camp Gratitude

14 May

A delayed post about Calculus Camp 2010:

Camp has come and gone and they took their test last week. AP Calculus Year 2 is officially over. =(

Sa~d!

Lessons learned from this year:

  1. My “No complaining/ No sarcasm” rule was the best new rule this year. I didn’t see those getting to me last year but boy, they did! The sarcasms that used to be bearable and sometimes amusing in one class period was no longer funny up there for 3 straight days. This year, their attempts and their catches at how easily they complain was kinda cute actually, even despite my cold/fever and cold sores from the lack of sleep from the week prior.
  2. A responsibilities list was another amazing addition this year. Being terrified of the numbers doubling and being the only person planning everything AND remembering how many supplies I had to bring last year, I assigned one responsibility per person this year. For example, one person was in charge of my LCD projector, another was in charge of the speakers, three people for the snack boxes, one person for the smores supplies, etc etc. Whoever was leftover was assigned Wed night clean-up, Thurs night clean-up, and Friday morning cabin checks. Everyone had the list alongside the schedule so everyone knew what they were supposed to do. Awesome. I would’ve forgotten the entire box of Free Response booklets if it wasn’t for that list. =) Awesome.
  3. Gratitude is THE attitude. Last year as well as this year, we had to write a bunch of thank you cards to our donors at DonorsChoose.org for helping to support us. This year, we even had a bunch of teachers at our school to thank for supporting us. I think having them write this at the very end of camp and reflecting upon all the things we did that they were grateful for really ended the camp perfectly. They seemed to come back with nothing but the happy memories and none of the pains of the endless hours of studying. =) Gratitude is a HUGE motivator!!

(I swear I taught them the meaning of the saying properly… Maybe he took from it a different meaning that was useful for his life at the time??)

I do have the video that I had made them after camp and before their AP test, but I haven’t quite decided whether to put it online yet. I have permission slips for it, but I have first and last names in the video this time and it being 42 kids this year, I don’t think I want to take the chance… I do have beautiful kids you know. =)

Just know that it was better than last year’s video with more video footage (due to the Flip) and I even had an extras menus for extra pictures and the high ropes footage. =) That Flip is flippin awesome btw. I especially love how I can capture snapshots super easily in really great quality too! I even got the shy ones into the video this way and the Emo ones were caught smiling as well. What can I say? We were all laughs by the time camp was over. =)

Reciprocal Teaching Protocol

21 Apr

My Calculus class came in today and it was one of those homework nights last night where I know over half of them did not really get. We were finding volumes by cylindrical shells. So I did a little group work rotation thingy and it went so beautifully as it woke up all the burnt-out juniors and checked-out seniors.

  1. I assigned each group of 4 ONE homework problem with the answer. They were to draw a picture, solve it, show all the work, and make sure everyone in the group could explain it. They then wrote/drew the problem on one of the white boards all around the room. (this took about 20 minutes! oops? no wonder they had such a hard time on the hw?)
  2. Every pair from each group was then assigned either as Pair A or Pair B.
  3. For the next 15 minutes, Pair A would stand by their boards teaching those who come by with questions on their problem while all the Pair Bs walk around the class with their homework asking questions and getting them answered.
  4. Then, to allow everyone to get their questions answered, Pair B takes over teaching their problem while all the Pair As walk around.

It was seriously beautiful to watch, and though I don’t say it enough, I LOVE my white boards!! We set up a desk in front of each white board with 2 to 3 chairs and it was so cute watching these mini-classes take place. Over half of them didn’t actually sit, but would stand and point and ask questions.

I think I’m going to review Free Response problems this way for this class. Everyone does 6 problems, each group gets the rubric to ONE of the problems and becomes master of it, then they all rotate to grade the rest! Why didn’t I think of this sooner?!

Calculus Camp 2010!

26 Mar

Since this is on a math teacher’s site, here are the scores differently dissected, but all true:

  • 60% of my class passed last year
  • No one passed the year before
  • One person passed with a 3 the year before that (2007)
  • I had more 4s than 3s on that exam last year
  • Out of the Latinos only, pass rate was higher at 65%
  • Out of those that attended camp, pass rate was 71.4%

Here are some more numbers for this year

  • We have twice the number of students in AP Calc this year
  • We have half the budget as last year
  • We can have more pass this year
  • My secret hope is to beat the magnet scores this year

www.donorschoose.org/hwang

More true numbers?

  • 90% of my school are Latino
  • This is our first year out of PI after 7 years in
  • 40 out of my 42 students qualified for the test fee reduction b/c of their low family income
  • Most have them have never gone camping before. Never made a campfire, never gone on a hike.
  • All of them want to go to CALCULUS CAMP! =) b/c ALL of them are amazing!

Question Paddles

7 Mar

I REALLY like it when my students ask questions, don’t I. Question Cards were for collaboration within themselves, but these Question Paddles are mostly for ME when I am lecturing.

This year, my Calculus class was incredibly difficult to read. They were from a program called Advantaged Plus which expects a whole lot from them and I think somewhere along the line they got scared of asking questions. Or maybe I got scarier? Or maybe they’re harder to read because I don’t know them as well?

Anyways, I tried to think of what it was that they could be asking me, because a lot of the time they simply don’t know what to ask. They start to get so lost that they don’t know what they don’t know anymore!

So I came up with these paddles for them to hold up:

I figure there were really two main questions to ask if they were getting lost- either to repeat what I said because they didn’t get it, or to slow it down.

It actually took me awhile to figure out what they meant by “go slower.” Do I talk too fast? Do I write too fast? Then I understood. In Calculus, I just skip work and do the Algebra way too fast.

The green one I threw in there just for kicks. Get it? Green, yellow, and sort-of-red for GO, SLOW, or STOP.

The kids seemed to love the idea when they first walked in and saw them. Some of them just because they thought it was cute. They don’t use it as much as I’d like BUT I do like them for two main reasons:

  1. You know when you ask after a long lecture, “any questions”? and you get hit by the overwhelming silence , even if you knew it was crazy talk you just did? With these, I simply ask them to hold up a GREEN for if they got it and are ready to move on, the YELLOW if they sort of get it but not as comfortably as they’d like, or the RED if they really had idea what I just said.
    This does WONDERS for me as I can get a quick gauge of how they felt about the content. They can also see that they were not alone in a sea of red sometimes.
  2. During my lecture, these silence the hyper inquisitive ones and allow the shy ones to be heard silently. If anything, I just like the fact that it reminds them of the questions they could ask, even if its just TWO very simple ones. =P

Anyways, this was a very new problem this year and one that caught me off guard. Normally, especially like my freshmen class, they can’t seem to STOP asking me questions. I hope this works! How else can I get them to keep asking questions?

Michael’s Velocity Graph

2 Feb

Shoving aside yesterday’s terrible day of no one finishing their test AND everyone doing horribly on it, today’s problem was a lot of fun to work on.

Each pair got a small white board and marker. They drew their axes (Quad 1 & 4) with time vs. velocity. I then showed them the slide, explaining that the purple was just a number line and they had to graph Michael’s velocity.

I used a picture of MJ because of his famous moonwalk. The kids had kept saying “the particle moves backwards” on their responses so I wanted to see if they really knew what they were saying. Did they mean the particle was moving in the negative direction or doing the moonwalk?

This was just a simple slide show of MJ sliding across the number line, until about 5 seconds in, when the picture flips and he continues sliding to the right, but backwards.

Sure enough, I got graphs that looked like a sine curve instead of two hills. Mostly though, I got the graph of time vs x(t). Guess I’ve got more work cut out for me than I thought.

The actual powerpoint slide show is in the Box widget to the right. Included are some slides that my students had a hard time with a couple of tests ago in terms of reading velocity graphs.